Abstract
Heterogeneity in oxygen distribution in solid tumours is recognised as a limiting factor for therapeutic efficacy. Vessel normalisation strategies, aimed at rescuing abnormal tumour vascular phenotypes and alleviating hypoxia, have become an established therapeutic strategy. However, understanding of how pathological blood vessel networks and oxygen transport are related remains limited. In this paper, we establish a causal relationship between the abnormal vasculature of tumours and their heterogeneous tissue oxygenation. We obtain average vessel lengths and diameters from tumour allografts of three cancer cell lines and observe a substantial reduction in the ratio compared to physiological conditions. Mathematical modelling reveals that small values of the measured ratio λ (i.e. λ < 6) can bias haematocrit distribution in tumour vascular networks and drive highly heterogeneous tumour tissue oxygenation. Finally, we show an increase in the average λ value of tumour vascular networks following treatment with the DC101 anti-angiogenic cancer agent. Based on our findings, we propose a new oxygen normalisation mechanism associated with an increase in λ following treatment with antiangiogenic drugs.
1 Introduction
Tissue oxygenation plays a crucial role in the growth and response to treatment of cancer. Indeed, well-oxygenated tumour regions respond to radiotherapy better than hypoxic or oxygen-deficient regions, by up to a factor of three [1, 2]. Further, the increased rates of proteomic and genomic modifications and clonal selection associated with anoxia (i.e., total oxygen depletion), endow tumours with more aggressive and metastatic phenotypes [3, 4]. Heterogeneous oxygen distributions in solid tumours are commonly attributed to their abnormal vasculature [5, 6]. While vessel normalisation strategies, aimed at reducing tumour hypoxia [7], have been shown to improve survival in e.g. glioblastoma patients undergoing chemotherapy and/or radiotherapy [8], the identification of patients who will benefit from such combined treatments remains an open question [3].
Detailed, functional imaging of the tumour microenvironment would enable the development of patient-specific treatment plans [9]. However, the maximum spatial resolution for imaging hypoxia via Positron Emission Tomography (PET) is currently 3–5 millimetres [10]. This resolution is three orders of magnitude larger than the micrometre scale governing oxygen transport in tissue and cell responses to hypoxia [11]. Consequently, PET images effectively mask intratumoural heterogeneity [10], which can lead to poor outcomes and foster the emergence of resistant clones [12]. A mechanistic understanding linking abnormal tumour vascular structure at the micrometre scale and oxygen heterogeneity at the tissue-scale is currently lacking. In this paper, we show how a multidisciplinary approach, which combines imaging with mathematical and computational modelling can be used to close this resolution gap [11, 13].
Oxygen is transported through the vasculature by binding to haemoglobin in red blood cells (RBCs) [13]. Haematocrit, or the volume fraction of RBCs in whole blood, does not distribute uniformly at vessel bifurcations (i.e. branching points where three vessels meet) [14, 15]. At a bifurcation with one afferent and two efferent branches, it is typically assumed that the efferent branch with the highest flow rate will have the highest haematocrit [15, 16] due to, among other features, plasma-skimming caused by the presence of a RBC-depleted layer or cell free layer (CFL) [17]. Several theoretical models have been proposed to describe this effect e.g. [16, 18, 19]. Tumour vasculature is characterised by abnormal branching patterns, reduced average vessel lengths, and increased formation of arterio-venous shunts (see [20] for a review). While these changes can impact haematocrit splitting (HS), and tumour oxygenation, they have received little attention in the literature.
In this paper, we establish a causal relationship between the abnormal vasculature of tumours and their heterogeneous tissue oxygenation. When we extract average vessel lengths and diameters from tumour allografts of three cancer cell lines we observe a substantial reduction in compared to physiological conditions. Detailed numerical simulations describing the transport of RBCs in plasma reveal that the average measured λ value in the tumour allografts is too small for the CFL to recover full symmetry between consecutive branching points. Further, the resulting bias in haematocrit distribution propagates and amplifies across multiple branching points. We argue that this memory effect can explain observations of haemo-concentration/dilution in tumour vasculature [21] and well perfused vessels that are hypoxic [22].
Based on the RBC simulations, we propose a new haematocrit splitting rule that accounts for CFL disruption due to pathologically small λ values. We integrate this rule into existing models of tumour blood flow and oxygen transport [23] and observe a haematocrit memory effect in densely branched vessel networks. The predicted tissue oxygenation is highly heterogeneous and differs markedly from predictions generated using rules for haematocrit splitting specialised for healthy vessel networks (e.g. [24, 25, 26, 5, 27]). Finally, we show an increase in the average λ value of tumour vascular networks following treatment with the DC101 anti-angiogenic cancer agent. Based on our results, we postulate the existence of a previously unreported tumour oxygen normalisation mechanism associated with an increase in the λ value after treatment with anti-angiogenic drugs.
2 Results
2.1 Distance between vascular branching points is on average shorter in solid tumours than in healthy tissue
We implemented a protocol for in vivo imaging of tumour vasculature [28] and exploited our recently published methods for vessel segmentation [29, 30] and three-dimensional (3D) vascular network reconstruction to characterise the morphology of tumour vasculature (see Methods section for more details). Briefly, tumour allografts of three murine cancer cell lines (i.e. MC38, colorectal carcinoma; B16F10, melanoma; and LLC, Lewis lung carcinoma) were implanted in mice, controlled for size, and imaged through an abdominal window chamber using a multi-photon microscope over multiple days. The vascular networks in the 3D image stacks were segmented and the associated network skeletons and vessel diameters computed [29, 30]. Figure 1(a) shows the two-dimensional (2D) maximum projection of an example network dataset along with a 2D projection of its segmentation and a close-in overlaying segmentation and skeletonisation. Vessel lengths (L) and diameters (d) in the networks followed a right-skewed distribution resembling a log-normal distribution (Figure 1(b)-(d)). No correlation was found between the variables (Pearson’s r2 < 0.04 for all samples analysed, Figure 1(b)-(d), Supplementary Tables S2–S3).
Table 1 summarises last-day statistics for all the experiments and averages per cell line. In the example MC38 dataset from Figure 1(a), average vessel length and diameter were 143 μm and 45.5 μm, respectively We observe how the group average vessel length is 128.6 μm, 125.9 μm, 108.8 μm for MC38, B16F10, and LLC, respectively. The average diameters are 33 μm, 36.5 μm, 35.7 μm, respectively, which is within the range previously described for tumour vasculature [31]. In addition, the length-to-diameter ratios (λ) are 4.0, 3.4, 3.0, respectively, which is substantially smaller than typical λ values reported under physiological conditions in a variety of tissues (Table 2) and representative of the high branching density encountered in tumour vasculature [20].
2.2 Plasma skimming in tumour-like vasculature is biased by history effects arising from CFL dynamics
Our finding of reduced inter branching point distance in tumour tissue motivated us to investigate a potential causal relationship between the reduction in L and λ and the profoundly abnormal tumour haemodynamics and mass transport patterns described in the literature [40]. In particular, we are interested in unravelling potential haemorheological phenomena contributing to tumour heterogeneity and hypoxia.
The presence of a RBC-depleted region adjacent to the vessel walls (i.e. the cell free layer (CFL)) is a key contributor to plasma skimming (PS) [15, 16, 17]. Previous studies have shown CFL disruption after microvascular bifurcations and found that the length required for CFL recovery is in the region of 10 vessel diameters (d) for d < 40μm [16], 8 − 15d for d ∈ [20, 24]μm [41], and 25d for d ∈ [10, 100]μm [42]. These values are substantially higher than the average λ values given in Table 1 and therefore we expect that, on average, CFL symmetry will not recover between the branching points in the networks under study.
Motivated by these findings, we exploited recent advances in blood flow simulation methods [43] to investigate the link between CFL dynamics and PS in a tumour-inspired microvascular network. Our intention is to understand whether CFL disruption effects arising at any given bifurcation can affect haematocrit splitting in downstream bifurcations for small inter-bifurcation distances relevant to tumour vasculature (see Methods section for further details). Briefly, we define a set of networks of cylindrical channels of constant radius, consisting of one main channel with an inlet and an outlet at either side and two side branches, which effectively define two consecutive bifurcations (Supplementary Figure S1). We consider inter-bifurcation distances of four and 25 channel diameters based on our tumour vascular network analysis and the largest of the CFL recovery distances reviewed earlier. We position the two side branches on the same side of the main channel or on opposite sides. A computational model of liquid-filled deformable particles (discocytes approximating the shape of an RBC) suspended in an ambient fluid is used to simulate blood flow in the networks, with RBCs inserted at the network inlet and removed at the outlets (see Section 4 and Supplementary Material for a summary of the simulation parameters). Flow rates at the inlets and the outlets of the network are configured such that at each bifurcation flow is split evenly. We perform blood flow simulations (3 runs in each network, with random perturbations in the RBC insertion procedure) and, after the initial transient required to fully populate the network with RBCs, we quantify haematocrit by an RBC-counting procedure.
Figure 2a–2b and Table 3 show how haematocrit split is close to even at bifurcation 1 for all geometries studied, as would be predicted by existing HS theoretical models. However, different degrees of haematocrit splitting occur at bifurcation 2. In the doublet geometry, we observe haemodilution in branch 3 and haemoconcentration in branch 4 (16.8% vs 23%, p < 0.001, Figure 2b). We will refer to these as the unfavourable and favourable branches. These effects are no longer statistically significant in the same branches in the extended double-t geometry (19.1% vs 19.4%, p = 0.3, Figure 2a). The haemoconcentration/haemodilution effect is also present in the cross geometry but the branches experiencing it are interchanged (22.1% vs 17.1%, p < 0.001, Figure 2c). In contrast with these results, existing HS theoretical models would predict even haematocrit splitting at bifurcation 2, regardless of the inter-bifurcation distance, due to the prescribed symmetrical flow and geometry conditions.
On closer inspection, the dynamics of the CFL show how, after bifurcation 1, CFL width is initially negligible and rapidly increases on the side of channel 1 leading to the favourable branch (θ = 0, Figure 2f). Conversely, CFL width increases after the bifurcation and follows a downward trend in the opposite side (θ = π, Figure 2f). An inter-bifurcation distance of four diameters is not sufficient for the CFL width to equalise on both sides (Figures 2f). In contrast, CFL width has time to become symmetric on both sides for an inter-bifurcation distance of 25 diameters (Figure 2g).
Taken together, these results show how CFL asymmetry can cause uneven haematocrit split in bifurcation 2 (Figure 2e). Our results are consistent with the findings by Pries et al., describing how asymmetry of the haematocrit profile in the feeding vessel of a bifurcation has a significant influence on RBC distribution in the daughter vessels [16]. In addition, we provide quantitative evidence of how CFL asymmetry may be the main contributing factor.
Interestingly, we observe small but statistically significant asymmetries in the haematocrit split in bifurcation 1 in the extended double-t geometry (19.2% vs 20.8%, p < 0.001, Figure 2a) and cross geometry (19.7% vs 20.4%, p = 0.035, Figure 2c), which consistently favour the side branch. We attribute this secondary effect to an asymmetrical streamline split in the chosen geometry as investigated in [44].
We note that the effects described above depend on the angle defined by the planes containing the two consecutive bifurcations. Our data suggest that for an angle of radian the favourable/unfavourable effects will not be observed (Supplementary Figure S2).
2.3 Haematocrit history effects lead to highly heterogeneous oxygen distribution in solid tumours
Existing theoretical models of HS [45, 19, 46] do not reproduce the haemoconcentration/haemodilution effects in the previous section. We hypothesise that this is because they neglect CFL disruption at bifurcations and its impact on subsequent bifurcations. We propose a new haematocrit splitting (HS) model which accounts for CFL dynamics and show that it predicts history effects in dense networks (see the Methods section for details and Supplementary Material for a description of its validation using the results from Section 2.2). Computationally, the new model is significantly less expensive than the RBC simulations.
We use Murray’s law [47] and our experimentally measured values of λ to design an idealised vessel network (see Supplementary Figure S4a and Methods section for details). Most notably we choose equal flow split and radii in daughter branches of any bifurcation, a scenario where existing HS models would predict homogeneous haematocrit throughout the network. We simulate network blood flow using a Poiseuille flow approximation with a HS model originally proposed by Pries et al. [16, 45] (without memory effects) and our new model (accounting for memory effects). As for the RBC simulations, differences in haematocrit between daughter branches emerge after two bifurcations (Supplementary Figure S4c), and are amplified with increasing vessel generation number (Supplementary Figure S4d). By contrast, existing models predict uniform splitting of the flow and haematocrit if the daughter vessels have equal radii (Supplementary Figure S4d).
Our model predicts the emergence of a compensatory mechanism in daughter branches. Increased flow resistance in the branch experiencing haemoconcentration leads to partial re-routing of flow in the other branch (Supplementary Figure S4b). This, in turn, attenuates the haemoconcentration/haemodilution effects described in Section 2.2 due to HS dependence on flow ratios.
We now consider how this memory effect in the haematocrit distribution may affect oxygen distribution in the tissue being perfused by the network. Following [23] (see Methods section for a description of the coupled model), the calculated haematocrit distribution in the synthetic network acts as a distributed source term in a reaction-diffusion equation for tissue oxygen. We define sink terms so that oxygen is consumed at a constant rate everywhere within the tissue. The equation is solved numerically and oxygen distributions generated using the two HS models (with and without memory effects) are compared for a range of λ values. The results presented in Figure 3c and Supplementary Figure S5 show that for larger values of λ the differences in the oxygen distribution in the tissue for the two haematocrit splitting models are not statistically significant (λ = 10, p = 0.14). However, as λ decreases, statistically significant differences appear (for example, with λ = 4, p < 0.001). Without memory effects, the oxygen distributions become more focussed as λ decreases; with memory effects, the oxygen distributions are flatter for all values of λ. A similar trend is observed for moderate values of λ but the distributions are significantly more diffuse (see Supplementary Figure S5c).
2.4 Vascular normalisation therapies increase lambda ratio in tumours
Our findings of reduced λ ratio in tumour vasculature and associated predictions of increased oxygen heterogeneity led us to investigate whether existing vascular normalisation therapies modulate this parameter. Previous reports (Table 4) have extensively demonstrated in multiple animal models that anti-angiogenic treatment leads to reduction in tumour vessel diameters. In those studies that analyse vessel length and diameter posttreatment, vessel length either remains unchanged or decreases to a lesser extent than vessel diameter. These findings indicate an increase in lambda ratio post-treatment. Furthermore, Kamoun et al. also reported a reduction in tumour haemoconcentration post-treatement [48], which suggests an in vivo link between an increase in λ, haematocrit normalisation and oxygen transport homogenisation.
We validated these results in our animal model by calculating the λ ratio following DC101 treatment. Our results indicate that in the first two days post-treatment λ increases significantly and then decreases to match the control trend (Figure 4, Supplementary Table S1). This change is explained by a linear increase in vessel length immediately after treatment (absent in the control group), which is compensated after two days by an increase at a higher rate in vessel diameter (comparable to the control group) (Supplementary Figure S6).
3 Discussion
Hypoxia compromises the response of many tumours to treatments such as radiotherapy, chemotherapy and immunotherapy. Dominant causative factors for hypoxia associated with the structure and function of the tumour vasculature include tortuosity, immature blood vessels that are prone to collapse, and inadequate flow regulation. Motivated by morphological analyses of vascular networks from different tumour types and detailed computer simulations of RBC transport through synthetic networks, we have proposed a new, rheological mechanism for tumour hypoxia.
We analysed vascular networks from murine MC38, B16F10, and LLC tumour allografts. For each vessel segment within each network, we calculated a novel metric λ which is the ratio of its length and diameter. Average λ values for the three tumour cell lines were similar in magnitude (λ ∈ [3, 4.2]) and several fold smaller than values from a range of healthy tissues (λ ∈ [9.5, 70]).‘ Detailed numerical simulations of RBC transport in plasma confirmed previous reports of transient alterations in the CFL downstream of network bifurcations (e.g. asymmetries in the cross-sectional haematocrit profile following a bifurcation [54] and the temporal dynamics governing its axisymmetry recovery [42]). Further, for the λ values measured in our tumours and the capillary number considered in our simulations, the CFL did not become symmetric between consecutive branching points. This bias is amplified across branching points and drives haemoconcentration/haemodilution at the network level. Based on these findings, we developed a new rule for haematocrit splitting at vessel bifurcations that accounts for CFL disruption due to abnormally short vessel segments. We then used our existing computational software [23] to demonstrate that this haematocrit memory effect can generate heterogeneous oxygen distributions in tissues perfused by highly branched vascular networks and that the network metric λ controls the extent of this heterogeneity. Finally, we reported an increase in the average λ value of tumour vascular networks following treatment with the DC101 anti-angiogenic cancer agent.
The implications of our findings are multiple. We have introduced a simple metric to characterise tumour vasculature based on the mean length-to-diameter ratio of vessel segments (= λ), and demonstrated how it can generate oxygen heterogeneity in an idealised, densely vascularised, tissue model. Our findings, of structurally induced haemodilution in vascular networks with low λ values, provide a mechanistic explanation for experimental observations of haemodilution in tumour vascular networks [21], the existence of well-perfused vessels that are hypoxic [22], and a possible explanation for the presence of cycling hypoxia in tumour microenvironment [55]. We conclude that vessel perfusion is a poor surrogate for oxygenation in tissue perfused by vascular networks with low λ values. Further, predictions of tissue oxygenation based on diffusion-dominated oxygen transport (e.g. [24, 25, 26, 5, 27]) may be inaccurate if they neglect heterogeneity in the haematocrit distribution of the vessel network. One way to address the resolution gap in current imaging modalities is to leverage mathematical modelling to infer micrometre scale information about oxygen levels from tissue scale images [11]. Such a theoretical framework must account for the complex interplay between microvascular structure, blood rheology, and oxygen transport, as highlighted in the current work.
Finally, anti-angiogenic drugs have been shown to generate transient periods of heightened homogeneous tissue oxygenation, due to improved restructuring and reduced permeability of tumour vessels [52]. This phenomenon, termed ‘vascular normalisation’ [6], can correct the deficient transport capabilities of tumour vasculature, homogenise drug and oxygen coverage, and, thereby, improve radiotherapy and chemotherapy effectiveness [2]. Based on our findings, we postulate the existence of a previously unreported oxygen normalisation mechanism associated with an increase in the average λ value of tumour vascular networks post treatment with anti-angiogenic drugs. Our results demonstrate how such morphological changes would lead to a less heterogeneous haematocrit distribution and more uniform intratumoural oxygenation. Further experimental work, measuring haematocrit before and after anti-angiogenic treatment, is needed to test this hypothesis and elucidate its importance in comparison with established mechanisms of normalisation (e.g. permeability reduction, vessel decompression [56]). If confirmed, this finding would provide a theoretical foundation for the development of therapeutic approaches for the normalisation of tumour oxygenation involving the administration of vascular targeting agents that normalise λ and, therefore, homogenise haematocrit and tissue oxygenation. Possible mechanisms to be targeted would include, among others, the promotion of post-angiogenic vascular remodelling [57, 58, 59], in particular vessel pruning and diameter control, or the modulation of currently unexplored temporal regulators of vascular patterning [60].
In summary, tissue oxygenation is central to cancer therapy. Understanding what controls tumour tissue oxygen concentration and transport properties is key to improving the efficacy of cancer treatments based on new and existing methods. Unravelling the causal relationship between vessel network structure and tissue oxygenation will pave the way for new therapies.
4 Methods
4.1 Tumour allograft model and abdominal imaging window protocol
We used an abdominal window chamber model in mice, which allowed for intravital imaging of the tumors [28]. The abdominal window chamber was surgically implanted in transgenic mice on C57Bl/6 background that had expression of red fluorescent protein tdTomato only in endothelial cells. The murine colon adenocarcinoma - MC38, murine melanoma - B16F10, and murine Lewis Lung Carcinoma – LLC tumors with expression of green fluorescent protein (GFP) in the cytoplasm were induced by injecting 5 μl of dense cell suspension in a 50/50 mixture of saline and matrigel (Corning, NY, USA). For DC101 treatment, mice bearing MC38 tumors were treated with anti-mouse VEGFR2 antibody (clone DC101, 500 μg/dose, 27 mg/kg, BioXCell) injected intraperitoneally on the first and fourth day of imaging. Prior to imaging we intravenously injected 100 μl of Qtracker 705 Vascular Labels (Thermo Fisher Scientific, MA, USA) which is a blood-pool based labelling agent, thus allowing us to determine whether vessels were perfused or not. Isoflurane inhalation anesthesia was used throughout the imaging, mice were kept on a heated stage and in a heated chamber and their breathing rate was monitored. Tumor images were acquired with Zeiss LSM 880 microscope (Carl Zeiss AG), connected to a Mai-Tai tunable laser (Newport Spectra Physics). We used an excitation wavelength of 940 nm and the emitted light was collected with Gallium Arsenide Phosphide (GaAsP) detectors through a 524–546 nm bandpass filter for GFP and a 562.5–587.5 nm bandpass filter for tdTomato and with a multi-alkali PMT detector through a 670–760 bandpass filter for Qtracker 705. A 20x water immersion objective with NA of 1.0 was used to acquire a Zstacks-TileScan with dimensions of 512×512 pixels in x and y, and approximately 70 planes in z. Voxel size was 5 μm in the z direction and 0.83 μm × 0.83 μm in the x-y plane. Each tumor was covered by approximately 100 tiles. The morphological characteristics of tumor vasculature were obtained from the acquired images as previously described [29, 30]. All animal studies were performed in accordance with the Animals Scientific Procedures Act of 1986 (UK) and Committee on the Ethics of Animal Experiments of the University of Oxford.
4.2 RBC simulations in synthetic capillary networks
We define a set of networks of cylindrical channels of diameter d. An inlet channel of length 25d (channel 0) bifurcates into two channels of length δ and 25d at π and π/2 radians clockwise, respectively (channels 1 and 2). Channel 1 bifurcates into two channels of length 25d at π and α radians clockwise, respectively (channels 3 and 4). We consider the following configurations (Supplementary Figure S1): double-t geometry (δ = 4d, α = π/2), cross geometry (δ = 4d, α = 3π/2), and extended double-t geometry (δ = 25d, α = π/2).
A model of liquid-filled elastic membranes (discocytes of 8 μm diameter approximating the shape of an RBC) suspended in an ambient fluid is used to simulate blood flow in the networks. We use the fluid structure interaction (FSI) algorithm previously presented and validated by Krüger et al. [61], which is based on coupling the lattice Boltzmann method (LBM), finite element method (FEM), and immerse boundary method (IBM). The discocyte membranes are discretised into 500 triangles, which imposes a voxel size of 0.8μm on the regular grid used in the LBM simulation. The mechanical properties of the membrane are defined to achieve a capillary number (i.e. the ratio of viscous fluid stress acting on the membrane and a characteristic elastic membrane stress) of 0.1 in channel 0. The coupled algorithm is implemented in the HemeLB blood flow simulation software [62, 63] (http://ccs.chem.ucl.ac.uk/hemelb). Simulations ran on up to 456 cores of the ARCHER supercomputer taking 11–32 hours. See Supplementary Material A.2 for full details.
A constant flow rate of and a procedure for RBC insertion with tube haematocrit Hinlet is imposed at the network inlet. The outlet flow rates are set to Q2= Q0/2 and Q3,4 = Q0/4 to ensure an equal flow split at each bifurcation. RBCs are removed from the computational domain when they reach the end of any outlet channel. Table 5 summarises the key model parameters in the model. We performed blood flow simulations (3 runs in each network, with random perturbations in the RBC insertion procedure) and, after the initial transient required to fully populate the network with RBCs, we quantified haematocrit by an RBC-counting procedure.
4.3 Hybrid model for tissue oxygen perfusion that accounts for history effects in vascular networks
We first explain how our vascular networks are designed. Then, we describe how blood flow and haematocrit are determined. Next, we introduce the HS models and explain how CFL memory effects are incorporated and the resulting flow problem solved. We conclude by describing how the resulting haematocrit distribution is used to calculate oxygen perfusion in the surrounding tissue. The basic steps of our method are summarised in the flow chart in Supplementary Figure S3.
4.3.1 Network design
Our networks have one inlet vessel (with imposed blood pressure and haematocrit; we call this generation 0), which splits into two daughter vessels (generation 1), which then split into two daughter vessels (generation 2), and so on until a prescribed (finite) number of generations is reached. Thereafter, the vessels converge symmetrically in pairs until a single outlet vessel is obtained (with imposed blood pressure). At every bifurcation, the diameters of the two daughter vessels are assumed to be equal and determined by appealing to Murray’s law [47]. Using the same vessel diameters in all simulations, we vary vessel lengths, so that for all vessels in the network, the lengths equal the product of λ (which is fixed for a given network) and the vessel diameter. We focus on λ-values in the range measured in our tumours (see Section 2.1 and Supplementary Material).
4.3.2 Blood flow and haematocrit splitting
Network flow problem
Tissue oxygenation depends on the haematocrit distribution in the vessel network perfusing the tissue. The haematocrit distribution depends on the blood flow rates. These rates are determined by analogy with Ohm’s law for electric circuits, with the resistance to flow depending on the local haematocrit via the Fahraeus-Lindquist effect (for details, see Supplementary Material and [64]). The flow rates and haematocrit are coupled. We impose conservation of RBCs at all network nodes1. A HS rule must then be imposed at all diverging bifurcations.
HS model without memory effects
The empirical HS model proposed by Pries et al. [65] states that the volume fraction of RBCs entering a particular branch FQE depends on the fraction of the total blood flow entering that branch FQB as follows: where logit(x) = ln (x/(1 − x)), B serves to as a fitting parameter for the nonlinear relationship between FQE and FQB, and A introduces asymmetry between the daughter branches (note that for an equal flow split FQB = 0.5, A ≠ 0 yields uneven splitting of haematocrit). Finally, X0 is the minimum flow fraction needed for RBCs to enter a particular branch (for lower flow fractions, no RBCs will enter)2; the term (1 − 2X0) reflects the fact that the CFL exists in both daughter vessels (see Supplementary Figure S7a).
HS model with memory effects
We account for the effects of CFL disruption and recovery by modifying the parameters A and X0 (as already observed in [16]). For simplicity, and in the absence of suitable data, we assume that the parameter B is the same in both daughter branches. If X0,f (Af) and X0,u (Au) denote the values of X0 (A) in the favourable and unfavourable daughter branches (see Figure 2e), then our new model of HS can be written as: where subscripts f and u relate to favourable and unfavourable branches, respectively (see Supplementary Figure S7a for a graphical depiction). It is possible to rewrite Equation (2) in terms of the suspension flow rates Q ≡ QB and haematocrit levels H of the favourable f, unfavourable u, and parent P vessels as (for details see Supplementary Material):
This formulation of our HS model facilitates comparison with other HS models [18, 19, 46]. Functional forms for Af, X0,f and X0,u are based on our RBC simulation results and the existing literature (see Supplementary Material). We use an iterative scheme (as in [19]) to determine the flow rates and haematocrit in a given network.
4.3.3 Calculating the tissue oxygen distribution
We embed the vessel network (described in Section 4.3.1) in a rectangular tissue domain. A steady state reaction-diffusion equation models the tissue oxygen distribution, with source terms at vessel network locations proportional to the haematocrit there, and sink terms proportional to the local oxygen concentration modelling oxygen consumption by the tissue. This equation is solved numerically using Microvessel Chaste (see [23] and Supplementary Material for details). In order to highlight the influence of HS on tissue oxygen, we focus on the central 25% of the domain which is well-perfused and ignore the avascular corner regions (see Figures 3a and 3b)
Acknowledgments
We acknowledge Hugh Zachary Ford for the cartoon in Figure 2e, the contributions of the HemeLB development team, and Dr Costas D. Arvanitis for helpful comments on our manuscript. Software development was supported by the Engineering and Physical Sciences Research Council (EPSRC) (grant eCSE-001-010). Supercomputing time on the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) was provided by the “UK Consortium on Mesoscale Engineering Sciences (UKCOMES)” under the EPSRC Grant No. EP/R029598/1. T.K.’s and M.O.B’s contribution have been funded through two Chancellor’s Fellowship at The University of Edinburgh. M.O.B is supported by grants from EPSRC (EP/L00030X/1, EP/R021600/1), Fondation Leducq (17 CVD 03), and the European Union’s Horizon 2020 research and innovation programme under grant agreement No 801423. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007- 2013) under REA grant agreement No 625631 (obtained by BM). This work was also supported by Cancer Research UK (CR-UK) grant numbers C5255/A18085 and C5255/A15935, through the CRUK Oxford Centre. This work was supported by the Biotechnology and Biological Sciences Research Council UK Multi-Scale Biology Network, grant number BB/M025888/1. We would like to acknowledge funding from the UK Fluids Network (EPSRC grant number EP/N032861/1) for a Short Research Visit between the Edinburgh and Oxford teams.
Footnotes
↵3 Equally contributing authors
↵12 Equally contributing senior authors
↵1 , where we sum over all vessels i meeting at a given node with haematocrit Hi and signed flow rates (of magnitude Qi).
↵2 The dependences of A, B and X0 on the diameters of the participating vessels and on the parent vessel haematocrit are described in Supplementary Material, see equations (S.14)-(S.16).
↵3 Consistency of the model requires that Au = A − Ashiftf (l; dP).
References
- [1].↵
- [2].↵
- [3].↵
- [4].↵
- [5].↵
- [6].↵
- [7].↵
- [8].↵
- [9].↵
- [10].↵
- [11].↵
- [12].↵
- [13].↵
- [14].↵
- [15].↵
- [16].↵
- [17].↵
- [18].↵
- [19].↵
- [20].↵
- [21].↵
- [22].↵
- [23].↵
- [24].↵
- [25].↵
- [26].↵
- [27].↵
- [28].↵
- [29].↵
- [30].↵
- [31].↵
- [32].
- [33].
- [34].
- [35].
- [36].
- [37].
- [38].
- [39].
- [40].↵
- [41].↵
- [42].↵
- [43].↵
- [44].↵
- [45].↵
- [46].↵
- [47].↵
- [48].↵
- [49].
- [50].
- [51].
- [52].↵
- [53].
- [54].↵
- [55].↵
- [56].↵
- [57].↵
- [58].↵
- [59].↵
- [60].↵
- [61].↵
- [62].↵
- [63].↵
- [64].↵
- [65].↵
- [66].↵
- [67].↵
- [68].↵
- [69].↵
- [70].↵
- [71].↵
- [72].↵
- [73].↵
- [74].↵
- [75].
- [76].
- [77].